The envelope of the Ramsey oscillations is exponential indicating that dephasing is dominated by white noise. Under the assumption that this white noise component was due entirely to photon shot noise in the center resonator, we can infer the (residual) photon population \(\langle n \rangle \)  in the resonator in the weak dispersive regime (since \(\chi /\kappa < 1\) ); using \(\gamma ^{11} = 8\langle n \rangle \chi ^{2}/(\kappa + 4\chi ^{2}/\kappa )\)  [1]}, we obtain an average photon number \(\langle n \rangle \approx 0.02\)  consistent with thermal populations reported previously in other cQED setups. In the absence of correlated noise (for instance, \(1/f\) -type), the dephasing rate of the 1-2 transition can be approximated as \(\gamma _{\phi }^{12} \approx \gamma ^{22} + \gamma ^{11}\)  [2]}; this describes the dephasing of 1-2 Ramsey oscillations shown in Fig. REF .
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